# Geometric interpretation of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ $(\ell \geq 3)$

## TL;DR

This paper provides a geometric interpretation of the singularity properties of generalized distance-squared mappings from b2 to b3 and higher dimensions, explaining why they have exactly one or no singular points.

## Contribution

It offers a geometric explanation for the singularity behavior of these mappings depending on the target dimension b3 or higher.

## Key findings

- For b3-dimensional targets, mappings have exactly one singular point.
- For targets with dimension greater than 3, mappings have no singular points.
- The geometric interpretation clarifies the difference in singularity properties.

## Abstract

Generalized distance-squared mappings are quadratic mappings of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ of special type. In the case that matrices $A$ constructed by coefficients of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ ($\ell \geq3$) are full rank, the generalized distance-squared mappings having a generic central point have the following properties. In the case of $\ell=3$, they have only one singular point. On the other hand, in the case of $\ell>3$, they have no singular points. Hence, in this paper, the reason why in the case of $\ell=3$ (resp., in the case of $\ell>3$), they have only one singular point (resp., no singular points) is explained by giving a geometric interpretation to these phenomena.